Alternating Optimization Algorithms for PowerAdjustment and Receive Filter Design in Multihop

Wireless Sensor NetworksTong Wang, Rodrigo C. de Lamare, Senior Member, IEEE, and Anke Schmeink, Member, IEEE

Abstract—In this paper, we consider a multihop wireless sensornetwork with multiple relay nodes for each hop where theamplify-and-forward scheme is employed. We present algorith-mic strategies to jointly design linear receivers and the powerallocation parameters via an alternating optimization approachsubject to different power constraints which include global, localand individual ones. Two design criteria are considered: thefirst one minimizes the mean-square error and the second onemaximizes the sum-rate of the wireless sensor network. Wederive constrained minimum mean-square error and constrainedmaximum sum-rate expressions for the linear receivers and thepower allocation parameters that contain the optimal complexamplification coefficients for each relay node. An analysis of thecomputational complexity and the convergence of the algorithmsis also presented. Computer simulations show good performanceof our proposed methods in terms of bit error rate and sum-rate compared to the method with equal power allocation andan existing power allocation scheme.

Index Terms—Minimum mean-square error (MMSE) criterion,maximum sum-rate (MSR) criterion, power allocation, multihoptransmission, wireless sensor networks (WSNs), relays

I. INTRODUCTION

Recently, wireless sensor networks (WSNs) have attracteda great deal of research interest because of their uniquefeatures that allow a wide range of applications in the areas ofdefence, environment, health and home [1]. WSNs are usuallycomposed of a large number of densely deployed sensingdevices which can transmit their data to the desired destina-tion through multihop relays [2]. Considering the traditionalwireless networks such as cellular systems, the primary goal insuch systems is to provide high QoS and bandwidth efficiency.The base stations have easy access to the power supply andthe mobile user can replace or recharge exhausted batteries inthe handset [1]. However, power conservation is getting moreimportant, especially for WSNs. One of the most importantconstraints on WSNs is the low power consumption require-ment as sensor nodes carry limited, generally irreplaceable,power sources. Therefore, low complexity and high energyefficiency are the most important design characteristics for

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

T. Wang and A. Schmeink are with the Institute for Theoretical InformationTechnology, RWTH Aachen University, Germany.

R. C. de Lamare is with CETUC / PUC-Rio, Brazil and Department ofElectronics, University of York, U.K.

The work was originally funded by the UK MoD, the University of Yorkand was supported by DFG grant SCHM 2643/4-1 and the Alexander vonHumboldt Foundation

WSNs. In a cooperative WSN, nodes relay signals to eachother in order to propagate redundant copies of the samesignals to the destination nodes. Among the existing relayingschemes, the amplify-and-forward (AF) and the decode-and-forward (DF) are the most popular approaches [3], [4]. In theAF scheme, the relay nodes amplify the received signal andrebroadcast the amplified signals toward the destination nodes.In the DF scheme, the relay nodes first decode the receivedsignals and then regenerate new signals to the destinationnodes subsequently.

Some power allocation methods have been proposed forWSNs to obtain the best possible signal-to-noise ratio (SNR)or best possible quality of service (QoS) [5], [6] at the desti-nations. By adjusting appropriately the power levels used forthe links between the sources, the relays and the destinations,significant performance gains can be obtained for a givenpower budget. Most of the research on power allocation forWSNs are based on the assumption of perfect synchronizationand available channel state information (CSI) at each node. AWSN is said to have full CSI when all of its nodes haveaccess to accurate and up-to-date CSI. When full CSI isavailable to all the nodes, the power of each node can beoptimally allocated to improve the system efficiency and lowerthe outage probability [7] or bit error rate (BER) [8], [9].

In WSNs, some power allocation problems can be for-mulated as centralized or distributed optimization problemssubject to power constraints on certain groups of signals. Forthe centralized schemes [10], [11], a network controller isrequired which is responsible for monitoring the informationof the whole network such as the CSI and SNR, calculating theoptimum power allocation parameters of each link and sendingthem to all nodes via feedback channels. This scheme consid-ers all the available links but it has two major drawbacks. Thefirst one is the high computational burden and storage demandat the network controller. The second one is that it requiresa significant amount of control information provided by feed-back channels which leads to a loss in bandwidth efficiency.For the distributed schemes [12], each node only needs tohave the knowledge of its ’partner’ information and calculateits own power allocation parameter. Therefore, a distributedscheme requires less control information and is ideally suitedto WSNs. However, the performance of distributed schemes isinferior to centralized schemes [13].

Due to the inherent limitations in the sensor node size,power and cost [1], they are only able to communicatein a short range. Therefore, multihop communication [14]

is employed to enhance the coverage of WSNs. By usingmultihop transmissions, the rapid decay of the received signalwhich is caused by the increased transmission distance can beovercome. Moreover, pathways around the obstacles betweenthe source and destination can be provided to avoid the signalshadowing [15]. Several works about power allocation of mul-tihop transmission systems have been proposed in [16]-[20].The work reported in [16] develops a cross-layer model formultihop communication and analyzes the energy consump-tion of multihop topologies with equal distance and optimalnode spacing. Centralized and Distributed schemes for powerallocation are presented to minimize the total transmissionpower under a constraint on the BER at the destination in [17]and [18]. In [19], two optimal power allocation schemes areproposed to maximize the instantaneous received SNR undershort-term and long-term power constraints. In [20], the outageprobability is considered as the optimization criterion to derivethe optimal power allocation schemes under a given powerbudget for both regenerative and non-regenerative systems.

In this paper, we consider a general multihop WSN wherethe AF relaying scheme is employed. The proposed strategy isto jointly design the linear receivers and the power allocationparameters that contain the optimal complex amplificationcoefficients for each relay node via an alternating optimizationapproach [21], [22]. Two kinds of linear receivers are designed,the minimum mean-square error (MMSE) receiver and themaximum sum-rate (MSR) receiver. They can be consideredas solutions to constrained optimization problems where theobjective function is the mean-square error (MSE) cost func-tion or the sum-rate (SR) and the constraint is a bound on thepower levels among the relay nodes. Then, the constrainedMMSE or MSR expressions for the linear receiver and thepower allocation parameter can be derived. The major noveltyin these strategies presented here is that they are applicableto general multihop WSNs with multi source nodes and des-tination nodes, as opposed to the simple two-hop WSNs withone pair of source-destination nodes [5], [23], [24]. Unlike theprevious works on the power allocation for multihop systemsin [16]-[20], in our work, the power allocation and receivercoefficients are jointly optimized. The joint strategies wereproposed for a two-hop WSN with multiple relay nodes inour previous work [25]. In order to increase the applicabilityof our investigation, in this paper, we develop joint strategiesfor general multihop WSNs. They can be considered as anextension of the strategies proposed for the two-hop WSNsand more complex mathematical derivations are presented.Moreover, different kinds of power constraints can be con-sidered and compared. For the MMSE receiver, we presentthree strategies where the allocation of power level across therelay nodes is subject to global, local and individual powerconstraints. Another fundamental contribution of this work isthe derivation of a closed-form solution for the Lagrangianmultiplier (λ) that arises in the expressions of the powerallocation parameters. For the MSR receiver, the local powerconstraints are considered. We propose a strategy that employsiterations with the Generalized Rayleigh Quotient [27] tosolve the optimization problem in an alternating fashion. Somepreliminary results of these work have been reported in [26].

The main contributions of this paper can be summarized as:1) Constrained MMSE expressions for the design of

linear receivers and power allocation parameters formultihop WSNs. The constraints include the global,local and individual power constraints.

2) Constrained MSR expressions for the design of linearreceivers and power allocation parameters for multi-hop WSNs subject to local power constraints.

3) Alternating optimization algorithms that compute thelinear receivers and power allocation parameters in1) and 2) to minimize the mean-square error ormaximize the sum-rate of the WSN.

4) Analysis of the computational complexity and theconvergence of the proposed optimization algo-rithms.

The rest of this paper is organized as follows. Section IIdescribes the general multihop WSN system model. SectionIII develops three joint MMSE receiver design and power al-location strategies subject to three different power constraints.Section IV develops the joint MSR receiver design and powerallocation strategy subject to local power constraints. SectionV contains an analysis of the computational complexity andthe convergence. Section VI presents and discusses the sim-ulation results, while Section VII provides some concludingremarks.

II. MULTIHOP WSN SYSTEM MODEL

Consider a general m-hop wireless sensor network (WSN)with multiple parallel relay nodes for each hop, as shown inFig. 1. The WSN consists of N0 source nodes, Nm destinationnodes and Nr relay nodes which are separated into m − 1groups: N1, N2, ... , Nm−1. The index refers to the numberof nodes after a given phase of transmission starting with 0and going up to m−1. The proposed optimization algorithmsin this paper refer to a particular instance, for which the rolesof the nodes acted as sources, relays and destinations havebeen pre-detemined. In subsequent time slots these roles canbe swapped so that all nodes can actually work as potentialsources. We concentrate on a time division scheme withperfect synchronization, for which all signals are transmittedand received in separate time slots. The sources first broadcastthe N0×1 signal vector s which contains N0 signals in parallelto the first group of relay nodes. We consider an amplify-and-forward (AF) cooperation protocol in this paper. An extensionto other cooperation protocols is straightforward. Each groupof relay nodes receives the signals, amplifies and rebroadcaststhem to the next group of relay nodes (or the destinationnodes). In practice, we need to consider the constraints onthe transmission policy. For example, each transmitting nodewould transmit during only one phase. In our WSN system,we assume that each group of relay nodes transmits the signalto the nearest group of relay nodes (or the destination nodes)directly. We can use a block diagram to indicate the multihopWSN system as shown in Fig. 2. Let Hs denote the N1×N0

channel matrix between the source nodes and the first groupof relay nodes, Hd denote the Nm × Nm−1 channel matrixbetween the (m − 1)th group of relay nodes and destination

Source Nodes Destination NodesRelay Nodes

N0 N1 NmN2 Nm-1

Feedback

Channel

Fig. 1. An m-hop WSN with N0 source nodes, Nm destination nodes andNr relay nodes.

H1,2Hs Hd

s

v1 v2

x1 x2 xm-1d

F1A1 Fm-1Am-1

vd

Detector

Feedback

Channel

Q[ ]

A

Fig. 2. Block diagram of the multihop WSN system.

nodes, and Hi−1,i denote the Ni × Ni−1 channel matrixbetween two groups of relay nodes as described by

Hs =

hs,1

hs,2

...hs,N1

, Hd =

hm−1,1

hm−1,2

...hm−1,Nm

, Hi−1,i =

hi−1,1

hi−1,2

...hi−1,Ni

,

(1)where hs,j = [hs,j,1, hs,j,2, ..., hs,j,N0 ] for j = 1, 2, ..., N1

denote the channel coefficients between the source nodes andthe jth relay of the first group of relay nodes, hm−1,j =[hm−1,j,1, hm−1,j,2, ..., hm−1,j,Nm−1 ] for j = 1, 2, ..., Nm

denote the channel coefficients between the (m− 1)th groupof relay nodes and the jth destination node. Further, hi−1,j =[hi−1,j,1, hi−1,j,2, ..., hi−1,j,Ni−1 ] for j = 1, 2, ..., Ni denotesthe channel coefficients between the (i − 1)th group of relaynodes and the jth relay of the ith group of relay nodes. Thereceived signal at the ith group of relay nodes (xi) for eachphase can be expressed as:

Phase 1:x1 = Hss + v1, (2)

y1 = F1x1, (3)

Phase 2:x2 = H1,2A1y1 + v2, (4)

y2 = F2x2, (5)

...

Phase i: (i = 3, 4, ...,m− 1)

xi = Hi−1,iAi−1yi−1 + vi, (6)

yi = Fixi, (7)

At the destination nodes, the received signal can be ex-pressed as

d = HdAm−1ym−1 + vd, (8)

where v is a zero-mean circularly symmetric complex addi-tive white Gaussian noise (AWGN) vector with covariancematrix σ2I. The matrix Ai = diag{ai,1, ai,2, ..., ai,Ni} isa diagonal matrix whose elements represent the amplifica-tion coefficient of each relay of the ith group. The matrixFi = diag{E(|xi,1|2), E(|xi,2|2), ..., E(|xi,Ni |2)}−

12 denotes

the normalization matrix which can normalize the power of thereceived signal for each relay of the ith group of relays. Pleasenote that the property of the matrix vector multiplicationAy = Ya will be used in the next section, where Y is thediagonal matrix form of the vector y and a is the vector formof the diagonal matrix A. At the receiver, a linear detectoris considered where the optimal filter and optimal amplifi-cation coefficients are calculated. The optimal amplificationcoefficients are transmitted to the relays through the feedbackchannel. The block marked with a Q[·] represents a decisiondevice. In our proposed designs, the full CSI of the system isassumed to be known at all the destination nodes. In practice,a fusion center [28] which contains the destination nodes isresponsible for gathering the CSI, computing the optimal linearfilters and the optimal amplification coefficients. The fusioncenter also recovers the transmitted signal of the source nodesand transmits the optimal amplification coefficients to the relaynodes via a feedback channel.

III. PROPOSED JOINT MMSE DESIGN OF THE RECEIVERAND POWER ALLOCATION

In this section, three constrained optimization problems areproposed to describe the joint design of the linear receiver (W)and the power allocation parameter (a) subject to a global,local and individual power constraints. They impose differentpower limitations on all the relay nodes, each group of relaynodes and each relay node, respectively. The assumptionsof these power constraints could determine the degrees offreedom for allocating the power among the relay nodes whichwill affect the performance and the lifetime of the networks.

A. MMSE Design with a Global Power Constraint

We first consider the case where the total power of all therelay nodes is limited to PT . The proposed design can beformulated as the following optimization problem

[Wopt, a1,opt, ..., am−1,opt] = arg minW,a1,...,am−1

E[∥s − WHd∥2],

subject tom−1∑i=1

Pi = PT

(9)

where (·)H denotes the complex-conjugate (Hermitian) trans-pose, Pi is the transmitted power of the ith group of relay

nodes, and Pi = Ni+1aHi ai. In (9), we employ the equality asthe constraint other than the inequality because it is obviousthat the more available total power can be used the betterperformance can be achieved, which means that even if we setthe constraint to be an inequality, the best performance will beachieve when the power is set to be the maximum bound (i.e.PT ). In addition, for such optimization problems it is easierto derive algorithmic solutions for equality constraints ratherthan inequality constraints.

To solve this constrained optimization problem, we adopt analternating optimization approach whose global convergencehas been established in [21] and [22] so that the globalminimum value can be achieved. We modify the MSE costfunction using the method of Lagrange multipliers [29] whichyields the following Lagrangian function

L =E[∥s − WHd∥2] + λ(m−1∑i=1

Ni+1aHi ai − PT )

=E(sHs)− E(dHWs)− E(sHWHd) + E(dHWWHd)

+ λ(m−1∑i=1

Ni+1aHi ai − PT ).

(10)

By fixing a1, ..., am−1 and setting the gradient of L in (10)with respect to the conjugate of the filter W equal to zero,where (·)∗ denotes the complex-conjugate, we get

Wopt =[E(ddH)]−1E(dsH)

=[HdAm−1E(ym−1yHm−1)A

Hm−1HH

d + σ2nI]−1

× HdAm−1E(ym−1sH).

(11)

The optimal expression for am−1 is obtained by equating thepartial derivative of L with respect to a∗m−1 to zero

∂L∂a∗m−1

=− E(∂dH

∂a∗m−1

Ws) + E(∂dH

∂a∗m−1

WWHd)

+Nmλam−1

=− E(YHm−1HH

d Ws)+ E[YH

m−1HHd WWH(HdYm−1am−1 + vd)]

+Nmλam−1

=0.(12)

Therefore, we obtain

am−1,opt =[E(YHm−1HH

d WWHHdYm−1) +NmλI]−1

× E(YHm−1HH

d Ws)=[HH

d WWHHd ◦ E(ym−1yHm−1)

∗ +NmλI]−1

× [HHd W ◦ E(ym−1sH)∗u]

(13)

where ◦ denotes the Hadamard (element-wise) product andu = [1, 1, ..., 1]T .

Similarly, for i = 2, 3, ...m− 1, we have

∂L∂a∗i−1

= −E(∂dH

∂a∗i−1

Ws) + (∂dH

∂a∗i−1

WWHd) +Niλai−1

= 0(14)

where

∂dH

∂a∗i−1

= YHi−1

(m−1∏k=i

HHk−1,kFH

k AHk

)HH

d . (15)

Let

Bi−1 =m−1∏k=i

HHk−1,kFH

k AHk . (16)

Then, we get

ai−1,opt =[E(YHi−1Bi−1HH

d WWHHdBHi−1Yi−1) +NiλI]−1

× E(YHi−1Bi−1HH

d Ws)=[Bi−1HH

d WWHHdBHi−1 ◦ E(yi−1yHi−1)

∗ +NiλI]−1

× [Bi−1HHd W ◦ E(yi−1sH)∗u].

(17)

From (13) and (17), we conclude that

ai,opt =[E(YHi BiHH

d WWHHdBHi Yi) +Ni+1λI]−1

× E(YHi BiHH

d Ws)=[BiHH

d WWHHdBHi ◦ E(yiy

Hi )∗ +Ni+1λI]−1

× [BiHHd W ◦ E(yis

H)∗u]

(18)

where

Bi =

{ ∏m−1k=i+1 HH

k−1,kFHk AH

k , for i = 1, 2, ...,m− 2,I, for i = m− 1.

(19)Please see the Appendix to find the expressions of Fi,E(yiyHi ), and E(yisH). The expressions in (11) and (18)depend on each other. Thus, it is necessary to iterate themwith an initial value of ai (i = 1, 2, ...,m − 1) to obtain thesolutions.

The Lagrange multiplier λ can be determined by solvingm−1∑i=1

Ni+1aHi,optai,opt = PT . (20)

Letϕi = E(YH

i BiHHd WWHHdBH

i Yi) (21)

andzi = E(YH

i BiHHd Ws). (22)

Then, we get

ai = (ϕi +Ni+1λI)−1zi. (23)

When λ is a real value, we have

[(ϕi+Ni+1λI)−1]H = [(ϕi+Ni+1λI)H ]−1 = (ϕi+Ni+1λI)−1.(24)

Equation (20) becomesm−1∑i=1

Ni+1zHi (ϕi+Ni+1λI)−1(ϕi+Ni+1λI)−1zi = PT . (25)

Using an eigenvalue decomposition (EVD), we have

ϕi = QiΛiQ−1i (26)

where Λi = diag{αi,1, αi,2, ..., αi,Mi , 0, ..., 0} consists ofeigenvalues of ϕi and Mi = min{N0, Ni, Nm}. Then, weget

ϕi +Ni+1λI = Qi(Λi +Ni+1λI)Q−1i . (27)

Therefore, (25) can be expressed asm−1∑i=1

Ni+1zHi Qi(Λi +Ni+1λI)−2Q−1i zi = PT . (28)

Using the properties of the trace operation, (28) can be writtenas

m−1∑i=1

Ni+1tr((Λi +Ni+1λI)−2Q−1

i zizHi Qi

)= PT . (29)

Defining Ci = Q−1i zizHi Qi, (11) becomes

m−1∑i=1

Ni∑j=1

Ni+1(αi,j +Ni+1λ)−2Ci(j, j) = PT . (30)

Since ϕi is a matrix with at most rank Mi, only the first Mi

columns of Qi span the column space of E(YHi BiHH

d Ws)Hwhich causes the last (Ni−Mi) columns of zHi Qi to becomezero vectors and the last (Ni − Mi) diagonal elements ofCi are zero. Therefore, we obtain the {

∑m−1i=1 2Mi}th-order

polynomial in λ

m−1∑i=1

Mi∑j=1

Ni+1(αi,j +Ni+1λ)−2Ci(j, j) = PT . (31)

B. MMSE Design with Local Power Constraints

Secondly, we consider the case where the total power ofthe relay nodes in each group is limited to some value PT,i.The proposed method can be considered as the followingoptimization problem

[Wopt, a1,opt, ..., am−1,opt] = arg minW,a1,...,am−1

E[∥s − WHd∥2],

subject to Pi = PT,i, i = 1, 2, ...,m− 1,(32)

where Pi as defined above is the transmitted power of the ithgroup of relays, and Pi = Ni+1aHi ai. Using the method of La-grange multipliers again, we obtain the following Lagrangianfunction

L = E[∥s − WHd∥2] +m−1∑i=1

λi(Ni+1aHi ai − PT,i). (33)

Following the same steps as described in Section III.A, we getthe same optimal expression for W as in (11). The optimalexpression for the power allocation vector ai is different from

(18) and is given by

ai,opt =[BiHHd WWHHdBH

i ◦ E(yiyHi )∗ +Ni+1λiI]−1

× [BiHHd W ◦ E(yis

H)∗u],(34)

where

Bi =

{ ∏m−1k=i+1 HH

k−1,kFHk AH

k , for i = 1, 2, ...,m− 2,I, for i = m− 1.

(35)The Lagrange multiplier λi can be determined by solving

Ni+1aHi,optai,opt = PT,i i = 1, 2, ...,m− 1. (36)

Following the same steps as in Section III.A, we obtain (m−1){2Mi}th-order polynomials in λi

Mi∑j=1

Ni+1(αi,j+Ni+1λi)−2Ci(j, j) = PT,i, i = 1, 2, ...,m−1.

(37)

C. MMSE Design with Individual Power Constraints

Thirdly, we consider the case where the power of each relaynode is limited to some value PT,i,j . The proposed method canbe considered as the following optimization problem

[Wopt, a1,opt, ..., am−1,opt] = arg minW,a1,...,am−1

E[∥s − WHd∥2],

subject to Pi,j = PT,i,j , i = 1, 2, ...,m− 1, j = 1, 2, ..., Ni,(38)

where Pi,j is the transmitted power of the jth relay node inthe ith group, and Pi,j = Ni+1a

∗i,jai,j . Using the method

of Lagrange multipliers once again, we have the followingLagrangian function

L = E[∥s − WHd∥2] +m−1∑i=1

Ni∑j=1

λi,j(Ni+1a∗i,jai,j − PT,i,j).

(39)Following the same steps as described in Section III.A, we getthe same optimal expression for W as in (11), and the optimalexpression for the amplification coefficient

ai,j,opt = [ϕi(j, j) +Ni+1λi,j ]−1[zi(j)−

∑l∈I,l =j

ϕi(j, l)ai,l],

(40)where I = {1, 2, ..., Ni}, ϕi and zi have the same expressionas in (21) and (22). The Lagrange multiplier λi,j can bedetermined by solving

Ni+1a∗i,j,optai,j,opt = PT,i,j i = 1, 2, ...,m−1, j = 1, 2, ...Ni.

(41)Table I shows a summary of our proposed MMSE designswith global, local and individual power constraints which willbe used for the simulations. If the quasi-static fading channel(block fading) is considered in the simulations, we onlyneed two iterations. Alternatively, low-complexity adaptivealgorithms can be used to compute the linear receiver Wopt

and the power allocation parameter vector ai,opt.

TABLE ISUMMARY OF THE PROPOSED MMSE DESIGN WITH GLOBAL, LOCAL AND INDIVIDUAL POWER CONSTRAINTS

Global Power Constraint Local Power Constraints Individual Power ConstraintInitialize the algorithm by setting: Initialize the algorithm by setting: Initialize the algorithm by setting:

A =√

PT∑m−1i=1 NiNi+1

I Ai =√

PT,i

NiNi+1I for i = 1, 2, ...,m−1 ai,j =

√PT,i,j

Ni+1for i = 1, 2, ...,m− 1,

j = 1, 2, ..., Ni

For each iteration: For each iteration: For each iteration:1. Compute Wopt in (11). 1. Compute Wopt in (11). 1. Compute Wopt in (11).2. For i = 1, 2, ...,m− 1 2. For i = 1, 2, ...,m− 1 2. For i = 1, 2, ...,m− 1a) Compute ϕi and zi in (21) and (22). a) Compute ϕi and zi in (21) and (22). a) Compute ϕi and zi in (21) and (22).b) Calculate the EVD of ϕi in (26). b) Calculate the EVD of ϕi in (26). b) For j = 1, 2, ..., Ni

c) Solve λ in (31). c) Solve λi in (37). i) Solve λi,j in (41).d) Compute ai,opt in (18). d) Compute ai,opt in (34). ii) Compute ai,j,opt in (40).

IV. PROPOSED JOINT MAXIMUM SUM-RATE DESIGN OFTHE RECEIVER AND THE POWER ALLOCATION

In this section, we analyse the proposed joint MSR designof the receiver and the power allocation. By the MSR designs,the best possible SNR and QoS can be obtained at thedestinations. They will improve the spectrum efficiency whichis desirable for the WSNs with the limitation in the sensornode computational capacity. Only the local constraints areconsidered here, because of the MSR receiver we make useof the Generalized Rayleigh Quotient which is only suitableto solve optimization problems with vectors. It limits the typesof power constraints. By substituting (2)-(7) into (8), we get

d =C0,m−1s + C1,m−1v1 + C2,m−1v2+ ...+ Cm−1,m−1vm−1 + vd

=C0,m−1s +m−1∑i=1

Ci,m−1vi + vd,

(42)

where

Ci,j =

{ ∏jk=i Bk, if i 6 j,

I, if i > j,(43)

andB0 = Hs, (44)

Bi = Hi,i+1AiFi for i = 1, 2, ..., m− 2, (45)

Bm−1 = HdAm−1Fm−1. (46)

We focus on a system with one source node for simplicity. Thegeneralization to multiple sources amounts to performing theoptimization of the additional filters. Therefore, the expressionof the sum-rate (SR) in terms of bps/Hz for our m-hop WSNis expressed as

SR =1

mlog2

[1 +

σ2s

σ2n

wHC0,m−1CH0,m−1w

wH(∑m

i=1 Ci,m−1CHi,m−1)w

](bps/Hz),

(47)where w is the linear receiver, and (·)H denotes the complex-conjugate (Hermitian) transpose. Let

ϕ = C0,m−1CH0,m−1 (48)

and

Z =m∑i=1

Ci,m−1CHi,m−1. (49)

The expression for the sum-rate can be written as

SR =1

mlog2

(1 +

σ2s

σ2n

wHϕwwHZw

)=

1

mlog2(1 + ax), (50)

wherea =

σ2s

σ2n

(51)

and

x =wHϕwwHZw

. (52)

Since 1m log2(1 + ax) is a monotonically increasing function

of x (a > 0), the problem of maximizing the sum-rate isequivalent to maximizing x. In this section, we consider thecase where the total power of the relay nodes in each group islimited to some value PT,i (local constraints). The proposedmethod can be considered as the following optimization prob-lem:

[wopt, a1,opt, ..., am−1,opt] = arg maxw,a1,...,am−1

wHϕwwHZw

,

subject to Pi = PT,i, i = 1, 2, ...,m− 1

(53)

where Pi as defined above is the transmitted power of theith group of relays, and Pi = Ni+1aH

i ai. We note that theexpression wHϕw

wHZw in (53) is the Generalized Rayleigh Quotient.Thus, the optimal solution of our maximization problem canbe obtained: wopt is any eigenvector corresponding to thedominant eigenvalue of Z−1ϕ.

In order to obtain the optimal power allocation vector aopt,we rewrite wHϕw

wHZw and the expression is given by

wHϕwwHZw

=aHi Miai

aHi Piai + wHi Tiwi

, for i = 1, 2, ..., m−1, (54)

where

Mi =diag{wHi Ci+1,m−1Hi,i+1Fi}C0,i−1CH

0,i−1

× diag{FHi HH

i,i+1CHi+1,m−1wi},

(55)

Pi =diag{wHi Ci+1,m−1Hi,i+1Fi}(

i∑k=1

Ck,i−1CHk,i−1)

× diag{FHi HH

i,i+1CHi+1,m−1wi},

(56)

and

Ti =m∑

k=i+1

Ck,m−1CHk,m−1. (57)

Since the multiplication of any constant value and an eigen-vector is still an eigenvector of the matrix, we express thereceive filter as

wi =wopt√

wHoptTiwopt

. (58)

Hence, we obtain

wHi Tiwi = 1 =

Ni+1aHi aiPT,i

. (59)

By substituting (59) into (54), we get

wHϕwwHZw

=aHi Miai

aHi Niaifor i = 1, 2, ..., m− 1, (60)

whereNi = Pi +

Ni+1

PT,iI. (61)

Likewise, we note that the expression aHMiaaHi Niai

in (60) is theGeneralized Rayleigh Quotient. Thus, the optimal solution ofour maximization problem can be obtained: ai,opt is any eigen-vector corresponding to the dominant eigenvalue of N−1

i Mi

that satisfies aHi,optai,opt =PT,i

Ni+1. Here, the local power

constraints can be satisfied by employing a normalization.When considering the global power constraint PT , there isno unique solution of ai,opt (i = 1, 2, ...,m − 1) that satisfy∑m−1

i=1 Ni+1aHi,optai,opt = PT . Thus, for this reason, we donot consider the global power constraint. The solutions of wopt

and ai,opt depend on each other. Therefore it is necessary toiterate them with an initial value of ai (i = 1, 2, ...,m− 1) toobtain the optimum solutions.

In this section, two methods are employed to calculate thedominant eigenvectors. The first one is the QR algorithm[30] which calculates all the eigenvalues and eigenvectors ofa matrix. We can choose the dominant eigenvector amongthem. The second one is the power method [30] which onlycalculates the dominant eigenvector of a matrix. Hence, thecomputational complexity can be reduced. Table II shows asummary of our proposed MSR design with a local powerconstraint which will be used for the simulations. If thequasi-static fading channel (block fading) is considered in thesimulations, we only need two iterations.

V. ANALYSIS OF THE PROPOSED ALGORITHMS

In this section, an analysis of the computational complexityand the convergence of the algorithms is developed. Wefirst illustrate the computational complexity requirements ofthe proposed MMSE and MSR designs. We quantify thecomputational complexity of the algorithms, which require agiven number of arithmetic operations per iteration. The lower

TABLE IISUMMARY OF THE PROPOSED MSR DESIGN WITH LOCAL POWER

CONSTRAINTS

Initialize the algorithm by setting

Ai =√

PT,i

NiNi+1I for i = 1, 2, ...,m− 1

For each iteration:1. Compute ϕ and Z in (48) and (49).2. Using the QR algorithm or the power method to

compute the dominant eigenvector of Z−1ϕ,denoted as wopt.

3. For i = 1, 2, ...,m− 1a) Compute Mi and Ni in (55) and (61).b) Using the QR algorithm or the power method to

compute the dominant eigenvector of N−1i Mi,

denoted as ai.c) To ensure the local power constraint

aHi,optai,opt =PT,i

Ni+1, compute ai,opt =

√PT,i

Ni+1aHi aiai.

the number of operations the lower the power consumptionwill be. Then, we make use of the convergence results for thealternating optimization algorithms in [21], [22] and presenta set of sufficient conditions under which our proposed algo-rithms will converge to the optimal solutions.

A. Computational Complexity Analysis

Table III and Table IV list the computational complexity periteration in terms of the number of multiplications, additionsand divisions for our proposed joint linear receiver design(MMSE and MSR) and power allocation strategies. For thejoint MMSE designs, we use the QR algorithm to performthe eigendecomposition of the matrix. Please note that inthis paper the QR decomposition employs the Householdertransformation [30], [31]. The quantities nQ and nP denotethe number of iterations of the QR algorithm and the powermethod, respectively. For the computational complexity of λin Table III, it does not include the processing of solvingthe equation in (31), (37) and (41), because the method witha global power constraint, equation (31) is a higher orderpolynomial whose complexity is difficult to be quantified. Asthe multiplication dominates the computational complexity,in order to compare the computational complexity of ourproposed joint MMSE and MSR designs, the number ofmultiplications versus the number of relay nodes in each groupfor each iteration are displayed in Fig. 3 and Fig 4. For thepurpose of illustration, we set m = 3, N0 = 1, N3 = 2 andnQ = nP = 10. For the MMSE design, it can be seen thatour proposed receiver with a global constraint has the samecomplexity as the receiver with local constraints. In practice,when considering the processing of solving the equation in(31), (37), the method with a global constraint will require ahigher computational complexity than the local constraints andthe difference will become larger along with the increase of thenumber of hops (m). When the individual power constraintsare considered, the computational complexity is lower thanother constraints because there is no need to compute the

eigendecomposition for it. For the MSR design, employingthe power method to calculate the dominant eigenvectors hasa lower computational complexity than employing the QRalgorithm.

VI. ANALYSIS OF THE PROPOSED ALGORITHMS

In this section, an analysis of the computational complexityand the convergence of the algorithms is developed. We firstillustrate the computational complexity requirements of theproposed MMSE and MSR designs. Then, we make use of theconvergence results for the alternating optimization algorithmsin [21], [22] and present a set of sufficient conditions underwhich our proposed algorithms will converge to the optimalsolutions.

A. Computational Complexity Analysis

Table III and Table IV list the computational complexityper iteration in terms of the number of multiplications, ad-ditions and divisions for our proposed joint linear receiverdesign (MMSE and MSR) and power allocation strategies.For the joint MMSE designs, we use the QR algorithm toperform the eigendecomposition of the matrix. Please notethat in this paper the QR decomposition by the Householdertransformation [30], [31] is employed by the QR algorithms.The quantities nQ and nP denote the number of iterationsof the QR algorithm and the power method, respectively.For the computational complexity of λ in Table III, it doesnot include the processing of solving the equation in (31),(37) and (41), because of the method with a global powerconstraint, equation (31) is a higher order polynomial whosecomplexity is difficult to be summarized. As the multiplicationdominates the computational complexity, in order to comparethe computational complexity of our proposed joint MMSEand MSR designs, the number of multiplications versus thenumber of relay nodes in each group for each iteration aredisplayed in Fig. 3 and Fig 4. For the purpose of illustration,we set m = 3, N0 = 1, N3 = 2 and nQ = nP = 10. For theMMSE design, it can be seen that our proposed receiver with aglobal constraint has the same complexity as the receiver withlocal constraints. In practice, when considering the processingof solving the equation in (31), (37), the method with a globalconstraint will require a higher computational complexity thanthe local constraints and the difference will become largeralong with the increase of the number of hops (m). When theindividual power constraints are considered, the computationalcomplexity is lower than other constraints because there isno need to compute the eigendecomposition for it. For theMSR design, employing the power method to calculate thedominant eigenvectors has a lower computational complexitythan employing the QR algorithm.

B. Sufficient Conditions for Convergence

To obtain convergence conditions, we need to define ametric space and the Hausdorff distance that will extensivelybe used. A metric space is an ordered pair (M, d), where Mis a nonempty set, and d is a metric on M, i.e., a function

2 4 6 8 10 12 14 16 18 2010

2

103

104

105

106

N1=N

2

Num

ber

of M

ultip

licat

ions

Global ConstraintLocal ConstraintsIndividual Constraints

Fig. 3. Number of multiplications versus the number of relay nodes ofour proposed joint MMSE design of the receiver and the power allocationstrategies.

2 4 6 8 10 12 14 16 18 2010

2

103

104

105

106

N1 = N

2

Num

ber

of M

ultip

licat

ions

Local Constraints, QR AlgorithmLocal Constraints, Power Method

Fig. 4. Number of multiplications versus the number of relay nodes of ourproposed joint MSR design of the receiver and the power allocation strategies.

d : M×M → R such that for any x, y, z ∈ M, the followingconditions hold:

1) d(x, y) ≥ 0.2) d(x, y) = 0 iff x = y.3) d(x, y) = d(y, x).4) d(x, y) ≤ d(x, y) + d(y, z).The Hausdorff distance measures how far two subsets of a

metric space are from each other and is defined by

dH(X,Y ) = max

{supx∈X

infy∈Y

d(x, y), supy∈Y

infx∈X

d(x, y)

}.

(62)The proposed joint MMSE designs can be stated as an

alternating minimization strategy based on the MSE definedin (9) and expressed as

Wn ∈ arg minW∈Wn

MSE(W, ai,n−1), (63)

TABLE IIICOMPUTATIONAL COMPLEXITY PER ITERATION OF THE JOINT MMSE DESIGNS

Power Constraint Multiplications Additions Divisions

Nm(Nm − 1)(4Nm + 1)/6 Nm(Nm − 1)(4Nm + 1)/6+(N0 +Nm−1)N2

d +N2m−1Nm +(N0 +Nm−1)N2

m +N2m−1Nm

W All +N0Nm−1Nm +Nm−1Nm +N0Nm−1Nm −N2m + 2N0Nm Nm(3Nm − 1)/2

+∑m−1

i=2 {2N2i−1Ni +Ni−1N

2i +Nm−1Nm +Nm +

∑m−1i=2 {2Ni−1N

2i

+N0Ni−1Ni + 4Ni−1Ni + 2Ni} +N0(Ni−1 − 1)Ni −N2i +Ni}

∑m−1i=1 {nQ( 13

6N3

i + 32N2

i + 13Ni − 2)

∑m−1i=1 {nQ( 13

6N3

i −N2i − 1

6Ni + 1)

Global −N3i + 3N0N2

i +N0NiNi+1 +N2i } −N3

i + 3N0N2i +N0NiNi+1

∑m−1i=1 {nQ(Ni − 1)}

+∑m−2

i=1 {NiNi+1 +Ni+1} −N2i −N0Ni −Ni}∑m−1

i=1 {nQ( 136N3

i + 32N2

i + 13Ni − 2)

∑m−1i=1 {nQ( 13

6N3

i −N2i − 1

6Ni + 1)

λ Local −N3i + 3N0N2

i +N0NiNi+1 +N2i } −N3

i + 3N0N2i +N0NiNi+1

∑m−1i=1 {nQ(Ni − 1)}

+∑m−2

i=1 {NiNi+1 +Ni+1} −N2i −N0Ni −Ni}

Individual∑m−1

i=1 {N0N2i +N0NiNi+1 +N2

i +N0Ni}∑m−1

i=1 {N0N2i +N0NiNi+1 −N2

i −Ni}+

∑m−2i=1 {NiNi+1 +Ni+1}

Global∑m−1

i=1 {Ni(Ni − 1)(4Ni + 1)/6 +N2i + 1}

∑m−1i=1 {Ni(Ni − 1)(4Ni + 1)/6 +N2

i }∑m−1

i=1 {Ni(3Ni − 1)/2}

a Local∑m−1

i=1 {Ni(Ni − 1)(4Ni + 1)/6 +N2i + 1}

∑m−1i=1 {Ni(Ni − 1)(4Ni + 1)/6 +N2

i }∑m−1

i=1 {Ni(3Ni − 1)/2}

Individual 2∑m−1

i=1 Ni∑m−1

i=1 Ni∑m−1

i=1 Ni

TABLE IVCOMPUTATIONAL COMPLEXITY PER ITERATION OF THE JOINT MSR DESIGNS

Power Constraint Multiplications Additions Divisions

nQ( 136N3

m + 32N2

m + 13Nm − 2) nQ( 13

6N3

d −N2d − 1

6Nd + 1)

Local +Nm(Nm − 1)(4Nm + 1)/6 +N2m +N1Nm +Nm(Nm − 1)(4Nm + 1)/6 nQ(Nm − 1)

QR Algorithm +∑m−1

i=1 {NiN2m +NiNi+1 +Ni} −N2

m +N1Nm +∑m−1

i=1 NiN2m +Nm(3Nm − 1)/2

+∑m−1

i=2 {2N2i−1Ni +Ni−1N

2i +

∑m−1i=2 {2Ni−1N

2i

+Ni−1NiNm + 4Ni−1Ni + 2Ni} +Ni−1(Ni − 1)Nm −N2i +Ni}

wnPN2

m +Nm(Nm − 1)(4Nm + 1)/6 nPNm(Nm − 1)Local +N3

m +N2m +N1Nm +Nm(Nm − 1)(4Nm + 1)/6 +N3

m

Power Method +∑m−1

i=1 {NiN2m +NiNi+1 +Ni} −2N2

m +N1Nm +∑m−1

i=1 NiN2m Nm(3Nm − 1)/2

+∑m−1

i=2 {2N2i−1Ni +Ni−1N

2i +

∑m−1i=2 {2Ni−1N

2i

+Ni−1NiNm + 4Ni−1Ni + 2Ni} +Ni−1(Ni − 1)Nm −N2i +Ni}

∑m−1i=1 {nQ( 13

6N3

i + 32N2

i + 13Ni − 2)

∑m−1i=1 {nQ( 13

6N3

i −N2i − 1

6Ni + 1)

Local +Ni(Ni − 1)(4Ni + 1)/6 +Ni(Ni − 1)(4Ni + 1)/6 +NiNi+1∑m−1

i=1 {nQ(Ni − 1)

QR Algorithm +∑i

k=1 NkN2i + 3N2

i + 2NiNi+1 +Ni+1Nm −Ni+1 +Ni − 1} +Ni(3Ni − 1)/2}+Ni+1Nm + 3Ni + 2}+ 2N2

m +∑m−1

i=2 {∑i−1

k=1(Nk − 1)N2i +Nm +m− 1

a +N2i (i− 2) +Ni}+ 2N2

m − 2Nm∑m−1i=1 {nPN2

i

∑m−1i=1 {nPNr(Nr − 1)

Local +Ni(Ni − 1)(4Ni + 1)/6 +Ni(Ni − 1)(4Ni + 1)/6 +N3i

∑m−1i=1 {Ni(3Ni − 1)/2}

Power Method +∑i

k=1 NkN2i +N3

i + 3N2i + 2NiNi+1 −N2

i +NiNi+1 +Ni+1Nm −Ni+1 +Nm +m− 1

+Ni+1Nm + 3Ni + 2}+ 2N2m +Ni − 1}+

∑m−1i=2 {

∑i−1k=1(Nk − 1)N2

i+N2

i (i− 2) +Ni}+ 2N2m − 2Nm

ai,n ∈ arg mina∈ai,n

MSE(Wn, ai) for i = 1, 2, ...,m− 1 (64)

where the sets W, ai ⊂ M, and the sequences of compactsets {Wn}n≥0 and {ai,n}n≥0 converge to the sets W and ai,respectively.

Although we are not given the sets W and ai directly, we

have the sequence of compact sets {Wn}n≥0 and {ai,n}n≥0.The aim of our proposed joint MMSE designs is to find asequence of Wn and ai,n(i = 1, 2, ...,m− 1) such that

limn→∞

MSE(Wn, a1,n, a2,n, ..., am−1,n)

= MSE(Wopt, a1,opt, a2,opt, ..., am−1,opt)(65)

where Wopt and ai,opt correspond to the optimal values ofWn and ai,n, respectively. Equation (65) can be considered asthe necessary condition of the following equations

limn→∞

MSE(Wn, ai,n) = MSE(Wopt, ai,opt)

for i = 1, 2, ...,m− 1(66)

if the other power allocation parameters aj,n(j = i) arekept constant when computing ai,n during the iterations. Topresent a set of sufficient conditions under which the proposedalgorithms converge, we need the so-called three-point andfour-point properties [21], [22]. Let us assume that there is afunction f : M×M → R such that the following conditionsare satisfied:

1) Three-point property (W, W, ai):For all n ≥ 1, W ∈ Wn, ai ∈ ai,n−1, andW ∈ argminW∈Wn

MSE(W, ai), we have

f(W, W) +MSE(W, ai) ≤ MSE(W, ai). (67)

2) Four-point property (W, ai, W, ai):For all n ≥ 1, W, W ∈ Wn, ai ∈ ai,n, andai ∈ argminai∈ai,n MSE(W, ai), we have

MSE(W, ai) ≤ MSE(W, ai) + f(W, W). (68)

These two properties are the mathematical expressions of thesufficient conditions for the convergence of the alternatingminimization algorithms which are stated in [21] and [22].It means that if there exists a function f(W, W) with theparameter W during two iterations that satisfies the twoinequalities for the MSE in (67) and (68), the convergence ofour proposed MMSE designs that make use of the alternatingminimization algorithm can be proved by the theorem below.

Theorem: Let {(Wn, ai,n)}n≥0, W, ai be compact subsectsof the compact metric space (M, d) such that

WndH→ W ai,n

dH→ ai (69)

and let MSE : M × M → R be a continuous function. Letconditions 1) and 2) hold. Then, for the proposed algorithms,we have

limn→∞

MSE(Wn, ai,n) = MSE(Wopt, ai,opt)

for i = 1, 2, ...,m− 1.(70)

Thus, equation (65) can be satisfied. A general proof of thistheorem is detailed in [21] and [22]. The proposed jointMSR designs can be stated as an alternating maximizationstrategy based on the SR defined in (47) that follows a similarprocedure to the one above.

VII. SIMULATIONS

In this section, we assess the performance of our proposedjoint designs of the linear receiver and power allocationmethods and compare them with the equal power allocationmethod which allocates the same transmitting power levelequally for all links from the relay nodes. For the purposeof fairness, we assume that the total transmitting power forall relay nodes in the network is the same which can be

indicated as∑m−1

i=1 PT,i =∑m−1

i=1

∑Ni

j=1 PT,i,j = PT . Weconsider a 3-hop (m=3) wireless sensor network as an exampleeven though the algorithms can be used with any numberof hops. The number of source nodes (N0), two groups ofrelay nodes (N1, N2) and destination nodes (N3) are 1, 4, 4and 2, respectively. We consider an AF cooperation protocol.The quasi-static fading channel (block fading channel) isconsidered in our simulations whose elements are Rayleighrandom variables (with zero mean and unit variance) andassumed to be invariant during the transmission of each packet.In our simulations, the channel is assumed to be knownat the destination nodes. For channel estimation algorithmsfor WSNs and other low-complexity parameter estimationalgorithms, one refers to [32] and [33]. During each phase,the sources transmit the QPSK modulated packets with 1500symbols. The noise at the destination nodes is modeled ascircularly symmetric complex Gaussian random variables withzero mean. A perfect (error free) feedback channel betweendestination nodes and relay nodes is assumed to transmit theamplification coefficients.

For the MMSE design, it can be seen from Fig. 5 thatour three proposed methods achieve a better performance thanthe equal power allocation method. Among them, the methodwith a global constraint has the best performance whereas themethod with individual constraints has the worst performance.This result is what we expect because a global constraintprovides the largest degrees of freedom for allocating thepower among the relay nodes whereas an individual constraintprovides the least. For the MSR design, it can be seen fromFig. 6 that our proposed method achieves a better sum-rateperformance than the equal power allocation method. Usingthe power method to calculate the dominant eigenvector yieldsa very similar result to the QR algorithm but requires a lowercomplexity.

0 1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

100

SNR (dB)

BE

R

Equal Power AllocationIndividual Constraints (Perfect Feedback Channel)Local Constraints (Perfect Feedback Channel)Global Constraint (Perfect Feedback Channel)Individual Constraints (BSC 8bits Pe=10e−3)Local Constraints (BSC 8bits Pe=10e−3)Global Constraint (BSC 8bit Pe=10e−3)

Fig. 5. BER performance versus SNR of our proposed joint MMSE designof the receiver and power allocation strategies, compared to the equal powerallocation method.

Besides the equal power allocation scheme, a MMSE powerallocation scheme reported in [34] where only the local powerconstraints are considered has also been used for comparison.

0 1 2 3 4 5 6 7 8 9 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Sum

−ra

te (

bps/

Hz)

Local Constraints, QA Algorithm (Perfect Feedback Channel)Local Constraints, Power Method (Perfect Feedback Channel)Local Constraints, QA Algorithm (BSC 8bits Pe=10e−3)Local Constraints, Power Method (BSC 8bits Pe=10e−3)Equal Power Allocation, QR AlgorithmEqual Power Allocation, Power Method

Fig. 6. Sum-rate performance versus SNR of our proposed joint MSR designof the receiver and power allocation strategies with local constraints, comparedto the equal power allocation method.

It can be seen from Fig. 7 that our proposed MMSE andMSR designs can achieve a very similar or better performance.Further advantage is that our proposed schemes only optimizethe relay amplifying vectors (or diagonal matrices) whereas in[34] the optimal relay amplifying matrices are needed whichrequires more feedback transmissions as well as informationexchanges among relay nodes in each group. Note that in orderto have a fair comparison, we only employ power allocationschemes for the relay nodes and assume every source nodehas unit transmitting power in the simulations.

0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Sum

−ra

te (

bps/

Hz)

MSR Design

0 5 1010

−3

10−2

10−1

100

SNR (dB)

BE

R

MMSE Design

Equal Power AllocationPower Allocation Scheme in [31]Proposed with Local Constraints

Fig. 7. (a) BER performance versus SNR of our proposed MMSE design (b)Sum-rate performance versus SNR of our proposed MSR design with localpower constraints and compare with the power allocation scheme in [31] andequal power allocation scheme.

In practice, the feedback channel cannot be error free. Inorder to study the impact of feedback channel errors on theperformance, we employ the binary symmetric channel (BSC)as the model for the feedback channel and quantize eachcomplex amplification coefficient to an 8-bit binary value (4bits for the real part, 4 bits for the imaginary part). The errorprobability (Pe) of the BSC is fixed at 10−3. The dashed

curves in Fig. 5 and Fig. 6 show the performance degradationcompared to the performance when using a perfect feedbackchannel. To show the performance tendency of the BSC forother values of Pe, we fix the SNR at 10 dB and choose Peranging from 0 to 10−2. The performance curves are shown inFig. 8 and Fig. 9 , which illustrate the BER and the sum-rateperformance versus Pe of our two proposed joint designs ofthe receivers. It can be seen that along with the increase inPe, their performance becomes worse.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0110

−3

10−2

10−1

Pe

BE

R

Individual Constraints (BSC 8Bits)Local Constraints (BSC 8Bits)Global Constraint (BSC 8Bits)

Fig. 8. BER performance versus Pe of our proposed MMSE designs whenemploying the BSC as the model for the feedback channel.

0 0.002 0.004 0.006 0.008 0.010.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pe

Sum

−ra

te (

bps/

Hz)

Local Constraints, QR Algorithm (BSC 8bits)Local Constraints, Power Method (BSC 8bits)

Fig. 9. Sum-rate performance versus Pe of our proposed MSR design whenemploying the BSC as the model for the feedback channel.

Finally, we replace the perfect CSI with the estimatedchannel coefficients to compute the receive filters and powerallocation parameters at the destinations. We employ theBEACON channel estimation which was proposed in [32].Fig. 10 illustrates the impact of the channel estimation onthe performance of our proposed MMSE and SMR designwith local constraints by comparing it to the performance ofperfect CSI. The quantity nt denotes the number of trainingsequence symbols per data packet. Please note that in these

0 5 1010

−3

10−2

10−1

100

SNR (dB)

BE

R

MMSE Design

BEACON, nt=10

BEACON, nt=50

Perfect CSI

0 5 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

Sum

−ra

te (

bps/

Hz)

MSR Design

BEACON, nt=10

BEACON, nt=50

Perfect CSI

Fig. 10. (a) BER performance versus SNR of our proposed MMSE design (b)Sum-rate performance versus SNR our proposed MSR design with local powerconstraints when employing the BEACON channel estimation, compared tothe performance of perfect CSI

simulations perfect feedback channel is considered and the QRalgorithm is used in the MSR design. For both the MMSE andMSR designs, it can be seen that when nt is set to 10, theBEACON channel estimation leads to an obvious performancedegradation compared to the perfect CSI. However, when nt isincreased to 50, the BEACON channel estimation can achievea similar performance to the perfect CSI. Other scenarios andnetwork topologies have been investigated and the results showthat the proposed algorithms work very well with channelestimation algorithms and a small number of training symbols.

VIII. CONCLUSIONS

In this paper, we have presented alternating optimizationalgorithms for receive filter design and power adjustmentwhich can be applied to general multihop WSNs. MMSEand MSR criteria have been considered in the developmentof the algorithmic solutions. Simulations have shown that ourproposed algorithms achieve a significant better performancethan the equal power allocation and power allocation schemein [34]. A possible extension of this work is employing low-complexity adaptive algorithms to compute the linear receiverand power allocation parameters. The algorithms can also beemployed in other multihop wireless networks along with non-linear receivers.

APPENDIX

Here, we derive the the expressions of Fi, E(yiyHi ), andE(yisH) that are used in Section II, III and VI. It holds that

Fi = diag{E(|xi,1|2), E(|xi,2|2), ..., E(|xi,Ni |2)}−12 (71)

where

E(|xi,j |2 =

σ2s |hs,j |2 + σ2

n, for i = 1,hi−1,jAi−1E(yi−1yHi−1)A

Hi−1hH

i−1,j + σ2n,

for i = 2, 3, ...,m,(72)

E(yiyHi ) =

Fi(σ

2sHsHH

s + σ2nI)FH

i , for i = 1,Fi[Hi−1,iAi−1E(yi−1yHi−1)A

Hi−1HH

i−1,i + σ2nI]FH

i

for i = 2, 3, ...,m,(73)

E(yisH) =

{σ2sFiHs, for i = 1,

FiHi−1,iAi−1E(yi−1sH), for i = 2, 3, ...,m.(74)

REFERENCES

[1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, ”ASurvey on Sensor Networks,” IEEE Commun. Mag., vol. 40, pp. 102-114, Aug. 2002.

[2] J. N. Laneman, D. N. C. Tse and G. W. Wornell, ”Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEETrans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004.

[3] Y. W. Hong, W. J. Huang, F. H. Chiu, and C. C. J. Kuo, ”CooperativeCommunications in Resource-Constrained Wireless Networks,” IEEESignal Process. Mag., vol. 24, pp. 47-57, May 2007.

[4] T. Peng, R. C. de Lamare and A. Schmeink, “Adaptive DistributedSpace-Time Coding Based on Adjustable Code Matrices for Coopera-tive MIMO Relaying Systems”, IEEE Trans. on Communications, vol.61, no.7, July 2013.

[5] T. Q. S. Quek, H. Shin, and M. Z. Win, ”Robust Wireless RelayNetworks: Slow Power Allocation With Guaranteed QoS,” IEEE J.Sel. Topics Signal Process., vol. 1, no. 4, pp. 700-713, Dec. 2007.

[6] K. Vardhe, D. Reynolds, and B. D. Woerner ”Joint Power Allocationand Relay Selection for Multiuser Cooperative Communication” IEEETrans. Wireless Commun., vol. 9, no. 4, pp. 1255-1260, Apr. 2010.

[7] Y. Li, B. Vucetic, Z. Zhou, and M. Dohler, ”Distributed Adaptive PowerAllocation for Wireless Relay Networks,” IEEE Trans. on WirelessCommun., vol. 6, no. 3, pp. 948-958, March 2007.

[8] R. C. de Lamare and S. Li, ”Joint Iterative Power Allocation and Lin-ear Interference Suppression Algorithms for Cooperative DS-CDMANetworks” in Proc. IEEE VTC, May 2010.

[9] R. C. de Lamare, “Joint iterative power allocation and linear inter-ference suppression algorithms for cooperative DS-CDMA networks”,IET Communications, vol. 6, no. 13, September 2012.

[10] M. Dohler, A. Gkelias, and H. Aghvami, ”Resource Allocation forFDMA-Based Regenerative Multihop Links,” IEEE Trans. on WirelessCommun., vol. 3, no. 6, Nov. 2004.

[11] J. Adeane, M. R. D. Rodrigues, and I. J. Wassell, ”Centralised anddistributed power allocation algorithms in cooperative networks” IEEE6th Workshop on Signal Processing Advances in Wireless Communica-tions, 2005.

[12] M. Chen, S. Serbetli, and A. Yener, ”Distributed Power AllocationStrategies for Parallel Relay Networks,” IEEE Trans. on WirelessCommun., vol. 7, no. 2, Feb. 2008.

[13] P. Clarke and R. C. de Lamare, “Transmit Diversity and Relay SelectionAlgorithms for Multirelay Cooperative MIMO Systems”, IEEE Trans.on Vehicular Technology, vol. 61 , no. 3, March 2012.

[14] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, ”Multihop Diversityin Wireless Relaying Channels,” IEEE Trans. on Commun., vol. 52,no. 10, pp. 1820-1830, Oct. 2004.

[15] H. Gharavi and K. Ban, ”Multihop sensor network design for wide-band communications,” Proceedings of the IEEE , vol. 91, no. 8, pp.1221-1234, Aug. 2003.

[16] Z. Shelby, C. Pomalaza-Raez, and J. Haapola, ”Energy Optimizationin Multihop Wireless Embedded and Sensor Networks,” in Proc. IEEEPIMRC, Sep. 2004.

[17] B. Maham and A. Hjorungnes, ”Near-Optimum Power Allocation forBER Restricted Multihop Cooperative Networks,” in Proc. IEEE ICC,May 2010.

[18] A. P. T. Lau and S. Cui, ”Joint Power Minimization in Wireless RelayChannels,” IEEE Trans. on Wireless Commun., vol. 6, no. 8, pp. 2820-2824, Aug. 2007.

[19] G. Farhadi and N. C. Beaulieu, ”Power-Optimized Amplify-and-Forward Multi-Hop Relaying Systems,” IEEE Trans. on WirelessCommun., vol. 8, no. 9, pp. 4634-4643, Sep. 2009.

[20] M. O. Hasna and M. Alouini, ”Optimal Power Allocation for RelayedTransmissions Over Rayleigh-Fading Channels,” IEEE Trans. on Wire-less Commun., vol. 3, no. 6, pp. 1999-2004, Nov. 2004.

[21] I. Csiszar and G. Tusnady, ”Information geometry and alternatingminimization procedure,” Statist. Decis.-Supplement Issue, no. 1, pp.205-237, 1984.

[22] U. Niesen, D. Shah, and G. W. Wornell, ”Adaptive alternating min-imization algorithms,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp.1423-1429, Mar. 2009.

[23] X. Deng and A. Haimovich, ”Power allocation for cooperative relayingin wireless networks,” IEEE Trans. Commun., vol. 50, no. 12, pp. 3062-3080, Jul. 2005.

[24] J. Huang, Z. Han, M. Chiang, and H. V. Poor, ”Auction-BasedResource Allocation for Cooperative Communications,” IEEE J. Sel.Areas Commun., vol. 26, no. 7, pp. 1226-1237, Sep. 2008.

[25] T. Wang, R. C. de Lamare, and A. Schmeink, ”Joint Linear ReceiverDesign and Power Allocation Using Alternating Optimization Algo-rithms for Wireless Sensor Networks,” IEEE Trans. Veh. Technol., vol.61, no. 9, Nov. 2012.

[26] T. Wang, R. C. de Lamare and A. Schmeink, ”Joint Maximum Sum-rate Receiver Design and Power Allocation Strategies for MultihopWireless Sensor Networks,” in Proc. IEEE ICASSP, Mar. 2012.

[27] R. D. Juday, ”Generalized Rayleigh quotient approach to filter opti-mization” J. Opt. Soc., vol. 15, no. 4, pp. 777-790, Apr. 1998.

[28] P. K. Varshney, Distributed Detection and Data Fusion, Springer-Verlag,1997.

[29] S. Haykin, Adaptive Filter Theory, 4th ed. Englewood Cliffs, NJ:Prentice-Hall, 2002.

[30] D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. JohnWiley & Sons, New York, 2002.

[31] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd ed.Baltimore, MD: Johns Hopkins Univ. Press, 1996.

[32] T. Wang, R. C. de Lamare, and P. D. Mitchell, ”Low-ComplexitySet-Membership Channel Estimation for Cooperative Wireless SensorNetworks,” IEEE Trans. Veh. Technol., vol. 60, no. 6, May, 2011.

[33] R. C. de Lamare and P. S. R. Diniz, ”Set-Membership Adaptive Algo-rithms based on Time-Varying Error Bounds for CDMA InterferenceSuppression,” IEEE Trans. Veh. Technol., vol. 58, no. 2, Feb. 2009.

[34] Y. Rong and Y Hua, ”Optimality of diagonalization of multi-hopMIMO relays,” IEEE Trans. on Wireless Commun., vol. 8, no. 12, pp.6068-6077, Dec. 2009.

Tong Wang received the B.Eng. degree in electricalengineering and automation from Beijing Universityof Aeronautics and Astronautics (BUAA), Beijing,China, in 2006, the M.Sc. degree (with distinction)in communications engineering and Ph.D. degreein electronic engineering from University of York,York, U.K., in 2008 and 2012, respectively.

He is currently a Research Associate with theInstitute for Theoretical Information Technology,RWTH Aachen University, Aachen, Germany. SinceJanuary 2014, he also works as a research fellow of

Alexander von Humboldt Foundation. His research interests include sensornetworks, cooperative communications, adaptive filtering, and resource opti-mization.

Rodrigo C. de Lamare (S’99 M’04 SM’10) re-ceived the Diploma in electronic engineering fromthe Federal University of Rio de Janeiro (UFRJ)in 1998 and the M.Sc. and Ph.D. degrees, both inelectrical engineering, from the Pontifical CatholicUniversity of Rio de Janeiro (PUC-Rio) in 2001 and2004, respectively.

Since January 2006, he has been with the Com-munications Research Group, Department of Elec-tronics, University of York, where he is a Professor.Since April 2012, he has also been a Professor at

PUC-RIO. His research interests lie in communications and signal processing,areas in which he has published over 300 papers in refereed journals andconferences.

Dr. de Lamare serves as an associate editor for the EURASIP Journal onWireless Communications and Networking and for IEEE Signal ProcessingLetters. He is a Senior Member of the IEEE and has served as the generalchair of the 7th IEEE International Symposium on Wireless CommunicationsSystems (ISWCS), held in York, U.K. in September 2010, and as the technicalprogramme chair of ISWCS 2013 held in Ilmenau, Germany in August 2013.

Anke Schmeink (M’09) received the Diploma de-gree in mathematics with a minor in medicine andthe Ph.D. degree in electrical engineering and infor-mation technology from RWTH Aachen University,Germany, in 2002 and 2006, respectively.

She worked as a research scientist for Philips Re-search before joining RWTH Aachen University in2008 where she is an associate professor since 2012.She spent several research visits with the Universityof Melbourne, Australia, and with the Universityof York, U.K. Anke Schmeink is a member of the

Young Academy at the North Rhine-Westphalia Academy of Science. Shehas received diverse awards, in particular, the Vodafone Young ScientistAward. Her research interests are in information theory, systematic designof communication systems and bioinspired signal processing.